Definability of the natural numbers in totally real towers of nested square roots

Authors:
Xavier Vidaux and Carlos R. Videla

Journal:
Proc. Amer. Math. Soc. **143** (2015), 4463-4477

MSC (2010):
Primary 03B25; Secondary 11U05, 11R80

Published electronically:
March 18, 2015

MathSciNet review:
3373945

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Abstract | References | Similar Articles | Additional Information

Abstract: For the ring of integers of a totally real algebraic field, Julia Robinson defines a set such that either or it is an interval in . She then proves that if this set has a minimum, then the natural numbers can be defined in , and hence has undecidable first-order theory. All known examples are such that has a minimum which is either or . In this work, we construct two infinite families of subrings of such rings for which is strictly between and . In one family, the infimum is a minimum, whereas in the other family it is not, but we can still define the natural numbers in this case.

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Additional Information

**Xavier Vidaux**

Affiliation:
Departamento de Matemática, Universidad de Concepción, Casilla 160 C, Concepción, Chile

Email:
xvidaux@udec.cl

**Carlos R. Videla**

Affiliation:
Department of Mathematics, Physics and Engineering, Mount Royal University, 4825 Mount Royal Gate SW, Calgary, Alberta, Canada T3E 6K6

Email:
cvidela@mtroyal.ca

DOI:
https://doi.org/10.1090/S0002-9939-2015-12592-0

Received by editor(s):
November 26, 2013

Received by editor(s) in revised form:
June 28, 2014

Published electronically:
March 18, 2015

Additional Notes:
Both authors have been supported by the first author’s Chilean research projects Fondecyt 1090233 and 1130134, and by Mount Royal University PD Funds.

Communicated by:
Mirna Džamonja

Article copyright:
© Copyright 2015
American Mathematical Society